Problem with singular Q matrix in discrete-time Riccati equations

Something like the following has to be solved for discrete-time LQR problems with a penalty on the output signal.

A = randn(2,2)
B = randn(2)
C = [-100 1]
Q = C'*C
dare(A, B, Q, 1)

However this code throws the error Q must be positive semi-definite.

The problem is that Q has a negative eigenvalue that is on the order of -eps(). I believe that this condition is too restrictive and a meaningful solution is found by both MatLab and MatrixEquations.jl for most A and B.

The numerics of RIccati equations might be a bit too much to get into, but this could be another reason to consider using MatrixEquations.jl (note that its ared function use the opposite order of the Q and R arguments).

A quick fix would be to just use something like Q = C'*C + eps()*I

Start by reproducing the discrete-time LQR example with A, B, C, Q, and dare shown in the issue, then inspect how dare validates Q. Compare the behavior with MATLAB and MatrixEquations.jl, accounting for the different argument order noted in the issue. Done means valid semidefinite Q inputs affected only by roundoff are accepted without the suggested eps()*I workaround.